We consider the set Irr(W) of (complex) irreducible characters of a finite Coxeter group W. The Kazhdan–Lusztig theory of cells gives rise to a partition of Irr(W) into “families” and to a natural partial order on these families. Following an idea of Spaltenstein, we show that can be characterised (and effectively computed) in terms of standard operations in the character ring of W. If, moreover, W is the Weyl group of an algebraic group G, then can be interpreted, via the Springer correspondence, in terms of the closure relation among the “special” unipotent classes of G.
Cite this article
Meinolf Geck, On the Kazhdan–Lusztig order on cells and families. Comment. Math. Helv. 87 (2012), no. 4, pp. 905–927DOI 10.4171/CMH/273